This chapter gives an extensive overview of various quaternionic geometries. The main focus is on positive quaternionic Kähler manifolds (orbifolds) and on hyper Kähler manifolds (orbifolds). Various ...
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This chapter gives an extensive overview of various quaternionic geometries. The main focus is on positive quaternionic Kähler manifolds (orbifolds) and on hyper Kähler manifolds (orbifolds). Various other quaternionic and hypercomplex geometries are introduced along the way. The hyper Kähler and quaternionic Kähler quotient construction is described. Other topics include the theory of toric hyper Kähler manifolds, the classification of positive toric self-dual and Einstein orbifolds, Hitchin's construction of SO(3)-invariant orbifold self-dual Einstein metrics on a 4-sphere, McKay's correspondence and Kronheimer's construction of ALE gravitational instantons.Less

Quaternionic Kähler and Hyperkähler Manifolds

Charles P. BoyerKrzysztof Galicki

Published in print: 2007-10-01

This chapter gives an extensive overview of various quaternionic geometries. The main focus is on positive quaternionic Kähler manifolds (orbifolds) and on hyper Kähler manifolds (orbifolds). Various other quaternionic and hypercomplex geometries are introduced along the way. The hyper Kähler and quaternionic Kähler quotient construction is described. Other topics include the theory of toric hyper Kähler manifolds, the classification of positive toric self-dual and Einstein orbifolds, Hitchin's construction of SO(3)-invariant orbifold self-dual Einstein metrics on a 4-sphere, McKay's correspondence and Kronheimer's construction of ALE gravitational instantons.

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. ...
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This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.Less

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

Claire Voisin

Published in print: 2014-02-23

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was ...
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This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.Less

On the Chow ring of K3 surfaces and hyper-Kahler manifolds

Claire Voisin

Published in print: 2014-02-23

This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ‎ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π‎ : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.